ARTICLE

Electromagnetic scattering laws in Weyl systemsMing Zhou1, Lei Ying1, Ling Lu2, Lei Shi3, Jian Zi3 & Zongfu Yu1

Wavelength determines the length scale of the cross section when electromagnetic waves

are scattered by an electrically small object. The cross section diverges for resonant scat-

tering, and diminishes for non-resonant scattering, when wavelength approaches infinity. This

scattering law explains the colour of the sky as well as the strength of a mobile phone signal.

We show that such wavelength scaling comes from the conical dispersion of free space at

zero frequency. Emerging Weyl systems, offering similar dispersion at non-zero frequencies,

lead to new laws of electromagnetic scattering that allow cross sections to be decoupled from

the wavelength limit. Diverging and diminishing cross sections can be realized at any target

wavelength in a Weyl system, providing the ability to tailor the strength of wave–matter

interactions for radiofrequency and optical applications.

DOI: 10.1038/s41467-017-01533-0 OPEN

1 Department of Electrical and Computer Engineering, University of Wisconsin, Madison, Madison, 53705, USA. 2 Institute of Physics, Chinese Academy ofSciences and Beijing National Laboratory for Condensed Matter Physics, Beijing, 100190, China. 3 Department of Physics, Fudan University, Shanghai,200433, China. Correspondence and requests for materials should be addressed to Z.Y. (email: zyu54@wisc.edu)

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1234

5678

90

E lectromagnetic scattering is a fundamental process thatoccurs when waves in a continuum interact with an elec-trically small scatterer. Scattering is weak under non-

resonant conditions; an example is Rayleigh scattering, which isresponsible for the colours of the sky. Conversely, scatteringbecomes much stronger with resonant scatterers, which have aninternal structure supporting localized standing waves, such asantennas, optical nanoresonators, and quantum dots. Resonantscatterers have wide application because the resonance allowsphysically small scatterers to capture wave energy from a largearea. As such, large electromagnetic cross sections, σ, are alwaysdesirable: a larger σ value means, for example, stronger mobilephone signals1 and higher absorption rates for solar cells2.

The maximum cross section of resonant scattering is boundedby the fundamental limit of electrodynamics. One might be

tempted to enlarge the scatterer to increase the cross section, butthis strategy only works for non-resonant scattering, or elec-trically large scatterers. In resonant scattering, physical size onlyaffects spectral bandwidth, while the limit of cross section isdetermined by the resonant wavelength λ as3:

σmax ¼Dπλ2 ð1Þ

The directivity D describes the anisotropy of the scattering; D = 1for isotropic scatterers. Equation 1 shows that an atom4 can havea σmax similar to that of an optical antenna5, despite the sub-nanometre size of the atom. This also means that optical scat-terers cannot attain resonant cross sections as large as those ofradiofrequency (RF) antennas, due to the smaller wavelengthsinvolved.

k k

Weyl point

Atom

RF antenna

1 1

!2 ∆!2∆!

Weyl pointDC

a b

c

!

! !

!=ck

" "

!

Fig. 1 The length scale of cross section and its relation to dispersion. a In free space, the resonant cross section scales according to σ $ λ2 or $ 1=ω2. Largecross sections always favour low frequencies. For example, the cross section of an optical transition in an atom is small (~10%12 m2) because of theassociated short wavelength (~μm). The cross section of an RF antenna is much larger (~10%4 m2) due to a much longer wavelength (~cm). Diverging crosssections are obtained around the DC point, which happens to be the apex of a conical dispersion. The double lines indicate double degeneracy due topolarization. b By embedding the resonant scatterer in a medium where the dispersion of the continuum exhibits conical dispersions located away from theDC point, diverging cross sections can be realized at high frequencies. The cross section scales according to $ 1=Δω2, where Δω is the relative detuningbetween the resonant frequency of the scatterer and the Weyl point. In both a and b, regions with stronger colour indicate larger resonant scattering crosssection. c Schematic of extraordinarily large cross section created by an electrically small resonant scatterer (red dot) placed inside Weyl photonic crystal

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Overcoming the limit just described has far-reaching implica-tions for RF and optoelectronic applications. Many efforts havebeen devoted to realizing this goal, including the use of enhanceddirectivity D6–10, degenerate resonances11, decreased dielectricconstants12 ϵ and materials with negative refractive index13.While these approaches exploit certain trade-offs to slightlyincrease the pre-factor in Eq. (1), the fundamental limit of λ2

remains, which can be proven directly from Maxwell’s equationswithout requiring specific scatterer details5. Until now, extremelylarge cross sections have only been obtained at long wavelengthsnear the DC frequency.

Here we show that the scattering laws in Weyl systems14 allowsthe cross section to be decoupled from the wavelength limit. Thisopens a new path to realizing strong wave–matter interaction,providing potential benefits to RF and optoelectronic devices thatrely on resonant scattering. Moreover, the scattering effect dis-covered here is equally applicable to acoustic or electronicwaves15–17.

ResultsLength scale of the cross section. In free space, the dispersiondirectly leads to the wavelength limit of the cross section, which isshown by Eq. (1). Specifically, the DC point, which is located atthe apex of the conical dispersion relation as shown in Fig. 1a,gives rise to the diverging cross sections at low frequencies. The

apex of the conic dispersion can be also realized at other spectralregimes, such as that shown in Fig. 1b. All the special scatteringproperties associated with the DC point can be reproduced,resulting in diverging cross sections and exceptionally stronglight–matter interactions at high frequencies. Weyl points14, 18–28,the three-dimensional (3D) analogy of Dirac points, have recentlybeen shown to exhibit such conical dispersion relations. Thescattering laws in Weyl systems allows the cross section to bedecoupled from the wavelength limit. Extraordinarily large crosssection can be realized even for very small scatterers inside a Weylphotonic crystal, which is schematically illustrated in Fig. 1c.Unlike the lensing effect that can concentrate incident waves to afixed focus, the scatterer concentrates incident waves to itself nomatter where it is placed inside the Weyl photonic crystal.

Conservation law of resonant scattering. To illustrate theunderlying physics of the relation between the dispersion andcross section, we will first show a conservation law of resonantscattering. We start by considering the resonant cross section of adipole antenna (Fig. 2a). Without losing generality, we only dis-cuss the scattering cross section, assuming zero absorption.Similar conclusions can be drawn for the absorptive case, with themaximum absorption cross section 1

4 that of the scattering crosssection29.

The dipole antenna is anisotropic due to its elongated shape, sothe cross section σðθ;φÞ depends on the incident direction of thewave. At a normal direction, when θ ¼ π=2, it reaches itsmaximum value5 of σmax ¼ 3λ2=2π. Along the axial direction,when θ ¼ 0, the cross section vanishes. While it is straightfor-ward to calculate the cross section of a dipole antenna, it is notimmediately apparent why the cross section follows the λ2 ruleand diverges around the DC frequency.

Figure 2b shows the real-space representation of σðθ;φÞ at theresonant frequency ω0. We can also represent σðθ;ϕÞ inmomentum space as σðkÞ, with k located on the isosurfacedefined by ω ¼ ω0. As shown in Fig. 2c, the isosurface is a spherewith jkj ¼ ω0=c. To visualize the momentum-space representa-tion, σðkÞ is indicated by the colour intensity on the isosurface inFig. 2d. As we show in Supplementary Note 1, the resonant crosssection satisfies the following conservation law:

∬s:ω kð Þ¼ω0

σ kð Þds ¼ 16π2 ð2Þ

The integration is performed on the isosurface ω kð Þ ¼ ω0. Ourproof is based on quantum electrodynamics30, so it applies toclassical scatterers such as antennas, as well as to quantumscatterers such as electronic transitions that absorb and emit light.More importantly, the continuum, in which the scatterer isembedded, does not need to be free space; it can be anisotropicmaterials, or even photonic crystals31, as long as a well-defineddispersion relation ω ¼ ωðkÞ exists.

Equation 2 dictates the scaling of σ with respect to the resonantfrequency ω0 of the scatterer. As ω0 decreases, the area of theisosurface shrinks. To conserve the value of the integration over asmaller isosurface, the cross section σ kð Þ must increase accord-ingly. For example, Fig. 2e and f show the isosurfaces of dipoleantennas with resonant frequencies at ω0=2 and ω0=3, respec-tively. The cross sections, indicated by colour intensity, mustincrease proportionally to maintain a constant integration overthese smaller isosurfaces. This can also be seen by the averagecross section in the momentum space:

σ &∬ σ kð Þds∬ ds

¼ 16π2

Sð3Þ

Real spacea b

c d

e f

z

yx

kz

ky

kx

#

Incident

"(k)ds

Momentum space

Isosurface

Dipoleantenna

k

$

"($, #)

"(k)/"d

"(k)

!0

!0/3!0/2

1.0

0.8

0.6

0.4

0.2

Fig. 2 Scaling law of resonant cross sections in momentum space. aSchematic of a dipole antenna in free space. b Real-space representation ofscattering cross section σ θ;φð Þ. c In the momentum space, the crosssection is represented by σðkÞ, with k located on the isosurface defined bythe resonant frequency. d The cross section σðkÞ of the dipole antenna isrepresented by the colour intensity on the isosurface. e The cross section ofa dipole antenna operating at a lower resonant frequency of ω0=2. Theisosurface shrinks by half compared to d. The cross section increases asindicated by stronger colours. f Same as e but with an even smallerresonant frequency of ω0=3. d, e and f all use the same colour map so thatthe cross section can be directly compared. The cross section is normalizedby σd ¼ 27λ20=2π

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Here, S & ∬ ds is the area of the isosurface. When approachingthe apex of the conical dispersion relation, the isosurfacediminishes, i.e., S ! 0. As a result, the cross section divergesaround the DC point (Fig. 1a).

Resonant scattering in Weyl photonic crystal. Recent demon-strations of Weyl points in photonic crystals18 show that thedispersion of a 3D continuum can exhibit conical dispersion atany designed frequency (Fig. 1b). The Hamiltonian for the con-tinuum around the Weyl point is given byH qð Þ ¼ vxqxσx þ vyqyσy þ vzqzσz , where σx;y;z are Pauli matrices.The momentum q qx; qy; qz

! "¼ k % kWeyl defines the distance to

the Weyl point in the momentum space, and q= 0 at the Weylpoint. The linear dispersion described by this Hamiltonian pro-duces an ellipsoidal isosurface that encloses the Weyl point. Theisosurface shrinks to a point at the Weyl frequency ωWeyl. Scat-tering properties associated with the DC point are carried to highfrequencies within Weyl medium, resulting in exceptionallystrong resonant scattering. As illustrated in Fig. 1b, the averagecross section scales as σ $ 1

ω0%ωWeylð Þ2 around the Weyl point. This

allows high frequency resonant scatterers, such as atoms andquantum dots, to attain large cross sections, potentially at amacroscopic scale.

We now demonstrate a specific example of resonant scatteringin a Weyl photonic crystal22. We consider a localized resonantscatterer in an infinitely large photonic crystal. The simulationsare performed in two steps. First, we numerically calculate theeigenmodes of the resonant frequency in the Brillouin zone byusing MIT photonic bands32 (MPB). Next, we use eacheigenmode as excitation and numerically calculate the scatteringcross section by using the quantum scattering theory wedeveloped recently33, which is described in detail in Supplemen-tary Note 2.

The structure of the Weyl photonic crystal consists of twogyroids, as shown in Fig. 3a. The magenta gyroid is defined by theequation f rð Þ>1:1, where f rð Þ ¼ sinð2πx=aÞ cosð2πy=aÞ þsinð2πy=aÞ cosð2πz=aÞ þ sinð2πz=aÞ cosð2πx=aÞ and a is thelattice constant. The yellow gyroid is the spatial inversion of themagenta one. The two gyroids are filled with a material with adielectric constant of ϵ ¼ 13. To obtain Weyl points, we add fourair spheres to the gyroids to break the inversion symmetry(Fig. 3a). These spheres are related by an S4ðzÞ transformation.The resulting band structure has four isolated Weyl points at thesame frequency of ωWeyl ¼ 0:5645 ð2πc=aÞ. Figure 3b illustratesconical dispersion in the 2kz ¼ %kx % ky plane.

The resonant scatterer is a quantum two-level system (TLS)embedded in the above photonic crystal. We numerically solvethe scattering problem using quantum electrodynamics33, 34. TheHamiltonian is ¼ Hpc þ HTLS þHI. The first two terms, Hpc ¼P

q !hωqcyqcq and HTLS ¼ !hω0byb, are the Hamiltonian of thephotons and the TLS, respectively35. Here, !h is the reducedPlanck constant, b† and b are the raising and lowering operatorsfor the quantum dot, respectively, and cyqand cq are the bosoniccreation and annihilation operators of the photons, respectively.The Lamb shift35 is incorporated into the resonant frequency ω0.The third term, HI ¼ i!h

Pq gqðcyqb% cqbyÞ, is the interaction

between the TLS and the radiation. The coupling coefficient isgq ¼ d ( Eq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω0=2!hϵ0L3

p, where d is the dipole moment, ϵ0 is

vacuum permittivity, Eq is the unit polarization vector of thephotons and L3 is the quantization volume.

Scattering inside photonic crystals is much more complex thanthat in free space. The continuum is highly dispersive, anisotropicand non-uniform, and thus the scattering cross section dependsstrongly on the location of the scatterer and orientation withinthe photonic crystal.

As an example, we consider a TLS with a transition frequencyω0 slightly below the Weyl frequency:ω0 % ωWeyl ¼ %0:0005 ð2πc=aÞ. The calculated cross sectionσ qð Þ (see Supplementary Note 2 for derivation) is plotted onthe isosurface as shown by Fig. 3e, f. As expected, it stronglydepends on the incident wavevector q. In addition, σ qð Þ variesgreatly at different locations, as shown by comparing Fig. 3e andFig. 3f. Despite all these differences, when integrated over theisosurface, ∬ σ qð Þds always results in the same constant: 16π2. Weperform the integration for a TLS at 20 different locations, allwith the same constant, as shown in Fig. 3d.

As the transition frequency of the TLS approaches the Weylpoint, i.e., ω0 ! ωWeyl, the isosurface shrinks in size, as illustratedby the insets of Fig. 4a. The conservation law leads to anincreasing σðqÞ, as shown by stronger colours. Near the Weylfrequency (black dashed line), the average cross sections σ isenhanced by three orders of magnitude compared to that in freespace, eventually diverging at the Weyl point (Fig. 4a). Theanalytical prediction from Eq. 2 and the area of the isosurfaceagree very well with predictions from numerical simulation(circles in Fig. 4a).

Frequency dependence of resonant scattering. A Weyl pointgreatly enhances the cross section at the resonant frequency.However, it comes at the price of suppressed cross section awayfrom the resonant frequency. Next, we discuss the spectral featureof the average cross section σðωÞ for a given TLS. The spectraldependence is shown in Supplementary Note 3 as

σ ωð Þ $ω0 % ωWeyl! "2

}

ω0 % ωð Þ2þ}2 ω0%ωWeylð Þ44

ð4Þ

Here, } is a constant that depends on the local electric field at theposition of the TLS, but does not vary significantly with frequency.In order to derive Eq. 4, we use the fact that the spontaneous decayrate is proportional to ω0 % ωWeyl

! "2 (Supplementary Eq. 32). Atthe resonance when ω0 % ω ¼ 0; the average cross section scales as1= ω0 % ωWeyl! "2, which increases as the resonant frequency moves

closer to the Weyl point. However, away from the resonancewhen jω0 % ωj ) jω0 % ωWeylj, Eq. 4 reduces toσ ω0ð Þ $ ω0 % ωWeyl

! "2= ω0 % ωð Þ2, which shows that being close

to the Weyl point suppresses the cross section. In Fig. 4b, we cal-culate the spectra for three different TLSs with their transitionfrequencies approaching the Weyl point (black dashed line). Whilethe peak value of the cross section grows, the full width at halfmaximum of the spectrum decreases. The spectral integration of thecross section remains around a constant (Supplementary Note 3).

Rayleigh or non-resonant scattering. The non-resonant scat-tering of electrically small objects in free space follows the Ray-leigh scattering law. In great contrast to resonant scattering, thecross section of Rayleigh scattering scales as σ $ ω4, anddiminishes at the DC frequency, as shown in Fig. 5a. Thisproperty is also closely related to the isosurface and can be carriedto high frequency at the Weyl point (see proof in SupplementaryNote 5). While the resonant cross section diverges, the non-resonant cross section diminishes at Weyl points, as illustrated inFig. 5b.

Using perturbation theory and the first-order Born approx-imation36, the Rayleigh scattering cross section can be shown as(details in Supplementary Note 5):

σ ω; kincð Þ ¼ ω2 ∬S:ω ksð Þ¼ω

d2ks uks r0ð ÞVukinc r0ð Þj j2/ ω2S ð5Þ

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Here, ukðrÞ is the eigenmode of the Weyl photonic crystalassociated with wavevector k. ks and kinc are the wavevectors ofthe scattered and incident eigenmodes, respectively. V is a tensorfor the scattering potential of the Rayleigh scatterer. The integralis proportional to the area of the isosurface S. At the DC point,the isosurface shrinks to a point with S ¼ 0, and the cross sectionof Rayleigh scattering is zero. Similarly, around the Weyl point,the area of the isosurface S $ Δω2 ¼ ω% ωWeyl

! "2. The cross

section scales as σ $ ω2Δω2, and is zero at the Weyl point(Fig. 5b).

To validate our theoretical prediction above, we numericallycalculate the Rayleigh scattering cross section of a small dielectricsphere embedded in the same Weyl photonic crystal. We useMPB32 to obtain the eigenmode of the Weyl photonic crystal,then numerically calculate the Rayleigh scattering cross sectionusing the normal-mode expansion37–39 (see more details in

x

y

z

!/ (

2%c/a)

kx /(2%/a) k y/(2%

/a)

a

c

b

d

"(q)/&2

16%2

A' (

B

C

0

500

250

00.005

–0.005

–0.005

0.005

0

0

0.005

11.0

0.57

0.568

0.566

0.564

0.562

0.56–0.2 –0.1 0 0.1 0.2 –0.2

0

0.2

Ez

0.5

–0.5

–1.0A B C A

0.00z

/a

x /a

–1

–1 0 1

–0.005

q z/(

2%/a)

q z/(

2%/a)

qx /(2%/a)

qx /(2%/a)q y

/(2%/a

)

q y/(2

%/a)

fe

00.005

–0.005

–0.005

0.005

0

0

0.005

–0.005

!!"(k)ds

Fig. 3 Simulation of resonant scattering in Weyl point photonic crystals. a The unit cell of the photonic crystal that supports Weyl points. Four air sphereswith a radius of 0.07a, where a is the lattice constant, are added to the gyroids to break the inversion symmetry. The positions of the air spheres are givenby ð 14 ; %

18 ;

12Þa,

14 ;

18 ; 0

! "a, ð58 ; 0;

14Þa and ð38 ;

12 ;

14Þa. b Band structure of the photonic crystal close to the Weyl points. The band structure is plotted on the

plane 2kz ¼ %kx % ky . Four Weyl points are created with conical dispersion relation. c The normalized electric-field distribution of one eigenmode of thephotonic crystal on the x–z plane. We plot the z-component of the electric field as an example. d The resonant cross sections of the two-level system (TLS)for different locations. The integration in the momentum space always leads to the same constant. The dipole moment of the TLS is in the x direction.Positions A-B-C-A are also labelled in c. e, f Examples of σðqÞ at two different positions α (e) and β (f) are plotted on the isosurfaces as colour intensity.The positions α and β are labelled in c. Both e and f use the same colour map so that the cross sections can be directly compared

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Supplementary Note 6). The calculated Rayleigh scattering crosssections around the Weyl point are obtained by averaging100 scatterers at random locations within one unit cell of thephotonic crystal, and are plotted in Fig. 5c as red circles. They arenormalized by σR, the scattering cross section of the samescatterer in free space. The blue dashed line indicates the scalinglaw σ $ ω2Δω2. The calculated Rayleigh scattering cross sectionagrees with the theoretical prediction well, with a vanishing crosssection observed at the Weyl point. For high-frequency devices,such as integrated waveguides and laser cavities, Rayleighscattering caused by interface roughness degrades performanceand increases noise. The combination of suppressed Rayleighscattering and enhanced resonant scattering could make Weylmedia attractive for these optoelectronic applications.

DiscussionMoreover, the conservation law of resonant scattering can also beextended to lower-dimensional space. In two-dimensional pho-tonic crystals, diverging cross sections can be realized at Diracpoints; an example is provided in Supplementary Note 4. Tofurther generalize the findings in this paper, we may not neces-sarily need conical dispersion. Quadratic dispersion found aroundthe band edges of photonic crystals also provides shrinking iso-surfaces. However, this is less useful in practice because the zerogroup velocity at the band edge makes it difficult to obtain pro-pagating waves40 in the presence of disorders. In addition, cou-pling into such media is difficult due to the large impedancemismatch.

As a final remark, the transport properties of electrons aroundDirac and Weyl points has also been studied in the past fewyears15–17. Some of the observations are consistent with thephysics of photon scattering shown in this paper. Here, weexplicitly show the general conservation law of cross section andits connection to the dispersion relation. We expect that similarconclusions can be drawn for both electrons and phonons. Itprovides useful insight to understand general scattering physicsbeyond Dirac and Weyl systems.

In conclusion, large resonant cross sections are of great prac-tical importance. They are only achievable with long resonantwavelengths when the frequency approaches the DC point. Asshown in this work, the dispersion, rather than the wavelength, is

responsible for the cross section. As a result, Weyl points, whichcan achieve similar conic dispersion as that around the DC point,lead to the diverging resonant cross section at any desired fre-quency. The exceptionally strong resonant scattering is also

SimulationAnalytical

!0/(2πc/a)

!0= 0.562 (2πc/a)

!0= 0.563 (2πc/a)

!0= 0.564 (2πc/a)"(q)/&2

a

Res

onan

t ave

rage

cro

ss s

ectio

n

b

Aver

age

cros

s se

ctio

n

104"0–

103"0–

103"0–

102"0–

10–1"0–

10–2"0–

10–3"0–

10–4"0–

10–5"0–

10–6"0

10"0–

"0–

102"0–

10"0–

"0–

0.560 0.562 0.564 0.566 0.568

500

400

300

200

100

0

0.570!0/(2πc/a)

0.559 0.560 0.561 0.562 0.563 0.564 0.565

Fig. 4 Resonant scattering cross section in Weyl photonic crystal. a Diverging resonant scattering cross section is realized around the Weyl frequency.Isosurfaces have an ellipsoidal shape (insets) with its colour indicating the value of the cross section; the shrinking isosurface leads to increasing crosssections around the Weyl frequency, which is indicated by the black dashed line. The results from quantum scattering simulation (red circles) agree wellwith the prediction based on the band structure (blue dashed line). The cross section is normalized by the average cross section in free space σ0 ¼ λ2=π. Itscales as σ $ 1=Δω2. b Spectra of average scattering cross sections for TLSs with different transition frequencies. For simplicity, the spontaneous emissionrates of the TLSs in vacuum are chosen to be 10%4ω0. We also assume the dipole moments and locations of the TLSs are the same. The black dashed linealso indicates the Weyl frequency

k kω=ck

Weyl point

Rayleighscattering

Weyl pointDC

a b

c

0.4"R

0.2"R

0

0.6"R

Aver

age

cros

s se

ctio

n

!/(2πc/a)0.560 0.562 0.564 0.566 0.568 0.570

!

!4!2∆!2

!

! !

" "

∆!

Fig. 5 Rayleigh scattering in free space and Weyl systems. a In free space, theRayleigh scattering cross section scales as σ $ ω4 and vanishes at the DCfrequency. b In Weyl systems, the Rayleigh scattering cross section scales asσ $ ω2Δω2 and vanishes at the Weyl frequency, which is indicated by theblack dashed line. In both a and b, regions with stronger red colours indicatesmaller Rayleigh scattering cross section. c Numerical calculation of Rayleighscattering in a Weyl photonic crystal. The Rayleigh scatterer is a dielectricsphere with a radius of 0.01a and a dielectric constant of 2. The numericalresults (red circles) are normalized by the Rayleigh scattering cross section infree space σR , and fit well the predictedσ $ ω2Δω2scaling (blue dashed line).The black dashed line also indicates the Weyl frequency

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accompanied by diminishing non-resonant scattering, which isalso similar to that around the DC point. Since Weyl points canbe realized at any frequency, we can effectively decouple the crosssection and the wavelength. It opens up possibilities for tailoringwave–matter interaction with extraordinary flexibility, which alsocan be extended to acoustic and electronic wave scattering.

Data availability. The data that support the finding of this studyare available from the corresponding author upon reasonablerequest.

Received: 25 May 2017 Accepted: 25 September 2017

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AcknowledgementsM.Z., L.Y. and Z.Y. acknowledge the finanical support of DARPA (YFA17 N66001-17-1-4049 and DETECT program). L.L. was supported by the National key R\&D Progam ofChina under Grant No. 2017YFA0303800, 2016YFA0302400 and supported by theNational Natural Science fundation of China (NSFC) under Project No. 11721404. L.S.and J.Z. were supported by 973 Program (2015CB659400), China National Key BasicResearch Program (2016YFA0301100) and NSFC (11404064).

Author contributionsM.Z. developed the theoretical formalism, performed the analytic calculations and per-formed the numerical simulations. All authors contributed to the final versions of themanuscript. Z.Y. supervised the project.

Additional informationSupplementary Information accompanies this paper at doi:10.1038/s41467-017-01533-0.

Competing interests: The authors declare no competing financial interests.

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© The Author(s) 2017

NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01533-0 ARTICLE

NATURE COMMUNICATIONS |8: �1388� |DOI: 10.1038/s41467-017-01533-0 |www.nature.com/naturecommunications 7

Supplementary Note 1: Conservation law of resonant scattering

As we discussed in the main text, the conservation law of resonant scattering applies to both classical and quantum resonances in any continuum with a well-defined dispersion relation. Here in this section, we will derive the conservation law using quantum electrodynamics.

To simplify the discussion, we consider a non-degenerate resonance embedded in a medium. The resonance frequency of the resonance is 𝜔𝜔0 . We also incorporate the Lamb shift into the resonance frequency to simplify the discussion1. The dispersion relation of the medium is defined by 𝜔𝜔(𝐤𝐤), where 𝐤𝐤 is the wave vector. The interaction between the resonance and the propagating waves in the medium is described by the interaction Hamiltonian 𝐻𝐻I. The exact expression of 𝐻𝐻I is not required to derive the conservation law as we will demonstrate below.

We consider the incident photon is in an eigenmode of the medium with a momentum of 𝐤𝐤i. The resonance is initially in its ground state |g⟩. The incident photon is scattered by the resonance to all the eigenmodes of the medium. For a final state with a momentum of 𝐤𝐤f, the transition probability is given by2

ℑ𝐟𝐟𝐟𝐟 =⟨g,𝐤𝐤𝐟𝐟|𝐻𝐻I|𝑒𝑒, 0⟩⟨𝑒𝑒, 0|𝐻𝐻I|g,𝐤𝐤i⟩

ℏ𝜔𝜔 − ℏ𝜔𝜔0 + 𝑖𝑖𝜋𝜋 ∑ |⟨𝑔𝑔,𝐤𝐤𝐟𝐟|𝐻𝐻I|𝑒𝑒, 0⟩|2𝛿𝛿�ℏ𝜔𝜔𝐤𝐤𝐟𝐟 − ℏ𝜔𝜔0�𝐤𝐤f (1)

where ℏ is the reduced Planck constant and |𝑒𝑒, 0⟩ indicates that the resonance is in its excited state and there is no photon in the medium. Using Fermi’s golden rule, we can obtain the optical cross section of the resonance as

𝜎𝜎(𝐤𝐤𝐟𝐟,𝜔𝜔) =1

𝑣𝑣𝐤𝐤𝐟𝐟 𝐿𝐿3⁄ �2πℏ

|ℑ𝐟𝐟𝐟𝐟|2𝛿𝛿�ℏ𝜔𝜔𝐤𝐤f − ℏ𝜔𝜔𝐤𝐤i�𝐤𝐤𝐟𝐟

(2)

The normalization factor 𝑣𝑣𝐤𝐤𝐟𝐟 𝐿𝐿3⁄ is the power flux of a single incident photon, where 𝑣𝑣𝐤𝐤𝐟𝐟 is the group

velocity of the incident photon and 𝐿𝐿3 is the quantization volume.

We focus on the resonant cross section at the resonant frequency 𝜔𝜔𝐤𝐤f = 𝜔𝜔𝐤𝐤i = 𝜔𝜔0. The summation in the momentum space then can be converted to an integration over the iso-surface S defined by 𝜔𝜔(𝐤𝐤) = 𝜔𝜔0 as

�𝛿𝛿�ℏ𝜔𝜔𝐤𝐤f − ℏ𝜔𝜔0�𝐤𝐤𝐟𝐟

= �𝐿𝐿

2𝜋𝜋�3�

𝑑𝑑𝑑𝑑ℏ𝑣𝑣𝐤𝐤𝐟𝐟𝑆𝑆

(3)

The resonant cross section at 𝜔𝜔𝐤𝐤f = 𝜔𝜔𝐤𝐤i = 𝜔𝜔0 then can be further calculated as

𝜎𝜎(𝐤𝐤𝐟𝐟,𝜔𝜔0) =2πℏ

|⟨𝑒𝑒, 0|𝐻𝐻I|𝑔𝑔,𝐤𝐤i⟩|2

𝑣𝑣𝐤𝐤𝐟𝐟 𝐿𝐿3⁄1

𝜋𝜋2 � 𝐿𝐿2𝜋𝜋�3∬ 𝑑𝑑𝑑𝑑

ℏ𝑣𝑣𝐤𝐤𝐟𝐟|⟨𝑔𝑔,𝐤𝐤𝐟𝐟|𝐻𝐻I|𝑒𝑒, 0⟩|2𝑆𝑆

= 16𝜋𝜋2|⟨𝑒𝑒, 0|𝐻𝐻I|𝑔𝑔,𝐤𝐤i⟩|2

ℏ𝑣𝑣𝐤𝐤𝐟𝐟∬ 𝑑𝑑𝑑𝑑

ℏ𝑣𝑣𝐤𝐤𝐟𝐟|⟨𝑔𝑔,𝐤𝐤𝐟𝐟|𝐻𝐻I|𝑒𝑒, 0⟩|2𝑆𝑆

(4)

Next, we integrate the resonant cross section 𝜎𝜎(𝐤𝐤𝐟𝐟,𝜔𝜔0) over the iso-surface S and immediately obtain the conservation law

�𝜎𝜎(𝐤𝐤𝐟𝐟,𝜔𝜔0)𝑑𝑑𝑑𝑑𝑆𝑆

= 16𝜋𝜋2∬ 𝑑𝑑𝑑𝑑

ℏ𝑣𝑣𝐤𝐤𝐟𝐟|⟨𝑒𝑒, 0|𝐻𝐻𝐼𝐼|𝑔𝑔,𝐤𝐤i⟩|2𝑆𝑆

∬ 𝑑𝑑𝑑𝑑ℏ𝑣𝑣𝒌𝒌𝒇𝒇

|⟨𝑔𝑔,𝐤𝐤𝐟𝐟|𝐻𝐻I|𝑒𝑒, 0⟩|2𝑆𝑆

= 16𝜋𝜋2 (5)

Supplementary Note 2: Quantum scattering theory

The resonant scattering cross section of a single two-level system (TLS) can be calculated using many approaches2,3. Here in our work, we calculated the scattering cross section and further numerically verified the conservation law by using the quantum scattering theory we recently developed4. In this section, we will describe our quantum scattering theory in detail.

As we described in the main text, the general Hamiltonian governing the interaction between photons and a single TLS is given by5

𝐻𝐻 = ℏ𝜔𝜔0𝑏𝑏†𝑏𝑏 +�ℏ𝜔𝜔𝐤𝐤𝑐𝑐𝐤𝐤†𝑐𝑐𝐤𝐤

𝐤𝐤

+ 𝑖𝑖ℏ�𝑔𝑔𝐤𝐤�𝑐𝑐𝐤𝐤†𝑏𝑏 − 𝑐𝑐𝐤𝐤𝑏𝑏†�

𝐤𝐤

(6)

Here 𝑏𝑏† (𝑏𝑏) is the raising (lowering) atomic operator of the TLS. The transition frequency of the TLS is 𝜔𝜔0. The creation (annihilation) operator for photons with angular frequency 𝜔𝜔𝐤𝐤 is 𝑐𝑐𝐤𝐤

†(𝑐𝑐𝐤𝐤), with 𝐤𝐤 being the momentum. The coupling strength between the TLS and the photon with momentum 𝐤𝐤 is given by 𝑔𝑔𝐤𝐤 = 𝐝𝐝 ∙ 𝐄𝐄�𝐤𝐤�𝜔𝜔0/2ℏ𝜖𝜖0𝐿𝐿3, where 𝐝𝐝 is the dipole moment of the TLS, 𝐄𝐄�𝐤𝐤 is the unit polarization vector of the photons at the location of the TLS and 𝜖𝜖0 is the vacuum permittivity.

Supplementary Figure 1: Quantum scattering theory. a Schematic of a two-level system in a continuum. Periodic boundary conditions with periodicity L are applied in both x and y directions. Because of the periodicity, incident photons can only be scattered to a set of discretized directions. b The 𝐪𝐪𝒎𝒎,𝒙𝒙𝒙𝒙 space is discretized by ∆𝑞𝑞 = 2π/𝐿𝐿, and represents a set of eigenmode channels (grey dots). c Real space bosonic creation operators. The real-space operators 𝑐𝑐F,𝑚𝑚,𝑛𝑛

† (𝜉𝜉) and 𝑐𝑐B,𝑚𝑚,𝑛𝑛† (𝜉𝜉) create a forward

and backward propagating single photon at position 𝜉𝜉 in the nth channel of the mth Weyl point, respectively. d Spatial single-photon wavefunctions 𝜓𝜓F,𝑚𝑚,𝑛𝑛(𝜉𝜉) and 𝜓𝜓B,𝑚𝑚,𝑛𝑛(𝜉𝜉) in the nth channel of the mth Weyl point. The coefficients 𝑟𝑟𝑚𝑚,𝑛𝑛 and 𝑡𝑡𝑚𝑚,𝑛𝑛 are the amplitudes of reflected and transmitted photons in the nth channel of the mth Weyl point, respectively.

The second term ∑ ℏ𝜔𝜔𝐤𝐤𝑐𝑐𝐤𝐤†𝑐𝑐𝐤𝐤𝐤𝐤 describes the propagating photons in the photonic crystals. Here we focus

on the Weyl-point photonic crystal. As we described in the main text, the dispersion is governed by the Weyl Hamiltonian ℋ(𝐪𝐪) = 𝑣𝑣𝑥𝑥𝑞𝑞𝑥𝑥𝜎𝜎𝑥𝑥 + 𝑣𝑣𝑦𝑦𝑞𝑞𝑦𝑦𝜎𝜎𝑦𝑦 + 𝑣𝑣𝑧𝑧𝑞𝑞𝑧𝑧𝜎𝜎𝑧𝑧, where 𝜎𝜎𝑥𝑥,𝑦𝑦,𝑧𝑧 are Pauli matrices. The momentum 𝐪𝐪�𝑞𝑞𝑥𝑥, 𝑞𝑞𝑦𝑦, 𝑞𝑞𝑧𝑧� = 𝐤𝐤 − 𝐤𝐤Weyl defines the distance to the Weyl point in the momentum space. Diagonalization of the Weyl Hamiltonian yields a three-dimensional (3D) conical dispersion around the frequency of the Weyl point 𝜔𝜔Weyl. To simplify the discussion, we consider isotropic Weyl points where 𝑣𝑣𝑥𝑥 = 𝑣𝑣𝑦𝑦 = 𝑣𝑣𝑧𝑧 = 𝑣𝑣. The dispersion around the Weyl point then can be simplified to 𝜔𝜔�𝐪𝐪 = 𝜔𝜔𝐤𝐤 − 𝜔𝜔Weyl =±𝑣𝑣|𝐪𝐪|. Note Weyl points typically appear in pairs and the minimum number of Weyl points is two. Here we assume the number of Weyl points is M and the position of mth Weyl point in the momentum space is

𝐤𝐤Weyl,𝑚𝑚 . By defining 𝐪𝐪𝑚𝑚 = 𝐤𝐤 − 𝐤𝐤Weyl,𝑚𝑚 , the Hamiltonian in Supplementary Equation (1) can be rewritten as

𝐻𝐻 = ℏ𝜔𝜔0𝑏𝑏†𝑏𝑏 + ��ℏ𝜔𝜔�𝐪𝐪𝑚𝑚𝑐𝑐𝐪𝐪𝑚𝑚† 𝑐𝑐𝐪𝐪𝑚𝑚

𝐪𝐪𝑚𝑚

𝑀𝑀

𝑚𝑚=1

+ 𝑖𝑖ℏ ��𝑔𝑔𝐪𝐪𝑚𝑚�𝑐𝑐𝐪𝐪𝑚𝑚† 𝑏𝑏 − 𝑐𝑐𝐪𝐪𝑚𝑚𝑏𝑏

†�𝐪𝐪𝑚𝑚

𝑀𝑀

𝑚𝑚=1

(7)

The key idea of our quantum scattering theory is to convert the summation over 𝐪𝐪𝑚𝑚 to a summation over a set of 1-dimension (1D) channels, or waveguides, where solutions have been successfully obtained6. Each channel carries a eigenmode of the photonic crystal with a distinct momentum 𝐪𝐪𝑚𝑚. They can be illustrated in the 𝐪𝐪𝑚𝑚-space as shown in Supplementary Figures 1a-b. Here, we use box quantization for the 𝐪𝐪𝑚𝑚 -space by setting up a periodic boundary condition in both x and y directions. At the end of derivation, we will take the periodicity L to ∞ to effectively remove the impact of this artificial periodic boundary condition. Because of the periodicity, incident photons can only be scattered to a set of discretized channels. These channels are defined by the in-plane wave vectors 𝐪𝐪𝑚𝑚,𝑥𝑥𝑦𝑦 of the wave, which are discretized by ∆𝑞𝑞 = 2𝜋𝜋/𝐿𝐿 (Supplementary Figure 1b). For a given frequency 𝜔𝜔𝐪𝐪𝑚𝑚, the channels are located on a sphere of radius |𝐪𝐪𝑚𝑚| = 𝜔𝜔𝐪𝐪𝑚𝑚/𝑣𝑣. Thereby, the total number of channels N is given by 𝑁𝑁 =∯ cos 𝜃𝜃𝐪𝐪𝑚𝑚 𝑑𝑑

2𝐪𝐪𝑚𝑚/(∆𝑞𝑞)2 /2 = 𝜋𝜋(𝜔𝜔�𝐪𝐪𝑚𝑚/𝑣𝑣∆𝑞𝑞)2. We can then convert the summation over 𝐪𝐪𝑚𝑚 to

�ℏ𝜔𝜔�𝐪𝐪𝑚𝑚𝑐𝑐𝐪𝐪𝑚𝑚† 𝑐𝑐𝐪𝐪𝑚𝑚

𝐪𝐪𝑚𝑚

= � � ℏ𝜔𝜔�𝑞𝑞𝑚𝑚,𝑛𝑛𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛† 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛

𝑞𝑞𝑚𝑚,𝑛𝑛

𝑁𝑁

𝑛𝑛=1

(8)

�𝑔𝑔𝐪𝐪𝑚𝑚�𝑐𝑐𝐪𝐪𝑚𝑚† 𝑏𝑏 − 𝑐𝑐𝐪𝐪𝑚𝑚𝑏𝑏

†�𝐪𝐪𝑚𝑚

= � �𝑔𝑔𝑞𝑞𝑚𝑚,𝑛𝑛�𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛† 𝑏𝑏 − 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛𝑏𝑏

†�𝑞𝑞𝑚𝑚,𝑛𝑛

𝑁𝑁

𝑛𝑛=1

(9)

Note the wave number 𝑞𝑞𝑚𝑚,𝑛𝑛 is a scalar. It’s also useful to differentiate waves with opposite group velocities as their dispersions are governed by two different branches. We define two scalar wave numbers 𝑞𝑞𝑚𝑚,𝑛𝑛F and 𝑞𝑞𝑚𝑚,𝑛𝑛

B for forward and backward propagating waves, respectively. The summations over 𝐪𝐪𝑚𝑚 then become

�ℏ𝜔𝜔�𝐪𝐪𝑚𝑚𝑐𝑐𝐪𝐪𝑚𝑚† 𝑐𝑐𝐪𝐪𝑚𝑚

𝐪𝐪𝑚𝑚

= �(� ℏ𝜔𝜔�𝑞𝑞𝑚𝑚,𝑛𝑛F 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛F† 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛F

𝑞𝑞𝑚𝑚,𝑛𝑛F

+ �ℏ𝜔𝜔�𝑞𝑞𝑚𝑚,𝑛𝑛B 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛B† 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛B

𝑞𝑞𝑚𝑚,𝑛𝑛B

)𝑁𝑁

𝑛𝑛=1

(10)

�𝑔𝑔𝐪𝐪𝑚𝑚�𝑐𝑐𝐪𝐪𝑚𝑚† 𝑏𝑏 − 𝑐𝑐𝐪𝐪𝑚𝑚𝑏𝑏

†�𝐪𝐪𝑚𝑚

= � �𝑔𝑔𝑞𝑞𝑚𝑚,𝑛𝑛F �𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛F† 𝑏𝑏 − 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛F 𝑏𝑏†�

𝑞𝑞𝑚𝑚,𝑛𝑛F

𝑁𝑁

𝑛𝑛=1

+� �𝑔𝑔𝑞𝑞𝑚𝑚,𝑛𝑛B �𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛B† 𝑏𝑏 − 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛B 𝑏𝑏†�

𝑞𝑞𝑚𝑚,𝑛𝑛B

𝑁𝑁

𝑛𝑛=1

(11)

where 𝜔𝜔�𝑞𝑞𝑚𝑚,𝑛𝑛F = 𝑣𝑣𝑞𝑞𝑚𝑚,𝑛𝑛F and 𝜔𝜔�𝑞𝑞𝑚𝑚,𝑛𝑛B = −𝑣𝑣𝑞𝑞𝑚𝑚,𝑛𝑛B are the dispersion relationships for forward and backward propagating photons in each channel, respectively. Note 𝑞𝑞𝑚𝑚,𝑛𝑛F and 𝑞𝑞𝑚𝑚,𝑛𝑛B are scalars with opposite signs.

Our next step is to convert the 𝐪𝐪𝑚𝑚 -space operators 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛F† and 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛B

† to real-space operators. It can be

realized by defining the following Fourier transformation

𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛F† = �1

𝐿𝐿� 𝑑𝑑𝜉𝜉𝑐𝑐F,𝑚𝑚,𝑛𝑛† (𝜉𝜉)𝑒𝑒𝑖𝑖𝑞𝑞𝑚𝑚F 𝜉𝜉

𝐿𝐿

−𝐿𝐿(12)

𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛B† = �1

𝐿𝐿� 𝑑𝑑𝜉𝜉𝑐𝑐B,𝑚𝑚,𝑛𝑛† (𝜉𝜉)𝑒𝑒𝑖𝑖𝑞𝑞𝑚𝑚B 𝜉𝜉

𝐿𝐿

−𝐿𝐿(13)

where operators 𝑐𝑐F,𝑚𝑚,𝑛𝑛† (𝜉𝜉) and 𝑐𝑐B,𝑚𝑚,𝑛𝑛

† (𝜉𝜉) create a forward and a backward propagating photon at position 𝜉𝜉 in the nth channel, respectively (Supplementary Figure 1c). By using rotating wave approximation, the summations over 𝑞𝑞𝑚𝑚,𝑛𝑛F further become

�ℏ𝜔𝜔�𝑞𝑞𝑚𝑚,𝑛𝑛F 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛F† 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛F

𝑞𝑞𝑚𝑚,𝑛𝑛F

= �ℏ𝑣𝑣𝑞𝑞𝑚𝑚,𝑛𝑛F

𝐿𝐿 � � 𝑑𝑑𝜉𝜉𝑑𝑑𝜉𝜉′∞

−∞

∞

−∞𝑐𝑐F,𝑚𝑚,𝑛𝑛† (𝜉𝜉)𝑐𝑐F,𝑚𝑚,𝑛𝑛(𝜉𝜉′)𝑒𝑒𝑖𝑖𝑞𝑞𝑚𝑚

F �𝜉𝜉−𝜉𝜉′�

𝑞𝑞𝑚𝑚,𝑛𝑛F

=ℏ𝑣𝑣∆𝑞𝑞𝐿𝐿� � 𝑑𝑑𝜉𝜉𝑑𝑑𝜉𝜉′

∞

−∞

∞

−∞𝑐𝑐F,𝑚𝑚,𝑛𝑛† (𝜉𝜉)𝑐𝑐F,𝑚𝑚,𝑛𝑛(𝜉𝜉′)� 𝑞𝑞𝑚𝑚,𝑛𝑛F 𝑒𝑒𝑖𝑖𝑞𝑞𝑚𝑚F �𝜉𝜉−𝜉𝜉′� 𝑑𝑑𝑞𝑞𝑚𝑚,𝑛𝑛F

∞

−∞

= 𝑖𝑖ℏ𝑣𝑣 � � 𝑑𝑑𝜉𝜉𝑑𝑑𝜉𝜉′∞

−∞

∞

−∞𝑐𝑐F,𝑚𝑚,𝑛𝑛† (𝜉𝜉)(−

𝑑𝑑𝑑𝑑𝜉𝜉)𝑐𝑐F,𝑚𝑚,𝑛𝑛

(𝜉𝜉′)𝛿𝛿(𝜉𝜉 − 𝜉𝜉′)

= 𝑖𝑖ℏ𝑣𝑣 � 𝑑𝑑𝜉𝜉𝑐𝑐F,𝑚𝑚,𝑛𝑛† (𝜉𝜉) �−

𝑑𝑑𝑑𝑑𝜉𝜉� 𝑐𝑐F,𝑚𝑚,𝑛𝑛

(𝜉𝜉)∞

−∞(14)

and

�𝑔𝑔𝑞𝑞𝑚𝑚,𝑛𝑛F �𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛F† 𝑏𝑏 − 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛F 𝑏𝑏†�

𝑞𝑞𝑚𝑚,𝑛𝑛F

= �𝑔𝑔𝑞𝑞𝑚𝑚,𝑛𝑛F �1𝐿𝐿 �� 𝑑𝑑𝜉𝜉𝑐𝑐F,𝑚𝑚,𝑛𝑛

† (𝜉𝜉)𝑒𝑒𝑖𝑖𝑞𝑞𝑚𝑚,𝑛𝑛F 𝜉𝜉𝑏𝑏∞

−∞− � 𝑑𝑑𝜉𝜉𝑐𝑐F,𝑚𝑚,𝑛𝑛(𝜉𝜉)𝑒𝑒−𝑖𝑖𝑞𝑞𝑚𝑚,𝑛𝑛F 𝜉𝜉𝑏𝑏†

∞

−∞�

𝑞𝑞𝑚𝑚,𝑛𝑛F

=𝑔𝑔𝑞𝑞𝑚𝑚,𝑛𝑛F

∆𝑞𝑞�1𝐿𝐿 �� 𝑑𝑑𝜉𝜉𝑐𝑐F,𝑚𝑚,𝑛𝑛

† (𝜉𝜉)𝑏𝑏∞

−∞� 𝑑𝑑𝑞𝑞𝑚𝑚,𝑛𝑛F 𝑒𝑒𝑖𝑖𝑞𝑞𝑚𝑚,𝑛𝑛F 𝜉𝜉∞

−∞− � 𝑑𝑑𝜉𝜉𝑐𝑐F,𝑚𝑚,𝑛𝑛(𝜉𝜉)𝑏𝑏†

∞

−∞� 𝑑𝑑𝑞𝑞𝑚𝑚,𝑛𝑛F 𝑒𝑒−𝑖𝑖𝑞𝑞𝑚𝑚,𝑛𝑛F 𝜉𝜉∞

−∞�

= 𝑔𝑔𝑚𝑚,𝑛𝑛√𝐿𝐿� 𝑑𝑑𝜉𝜉𝛿𝛿(𝜉𝜉)�𝑐𝑐F,𝑚𝑚,𝑛𝑛† (𝜉𝜉)𝑏𝑏 − 𝑐𝑐F,𝑚𝑚,𝑛𝑛(𝜉𝜉)𝑏𝑏†�

∞

−∞(15)

Similarly, the summations over 𝑞𝑞𝑚𝑚B further become

� ℏ𝜔𝜔�𝑞𝑞𝑚𝑚,𝑛𝑛B 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛B† 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛B

𝑞𝑞𝑚𝑚,𝑛𝑛B

= 𝑖𝑖ℏ𝑣𝑣 � 𝑑𝑑𝜉𝜉𝑐𝑐B,𝑚𝑚,𝑛𝑛† (𝜉𝜉) �−

𝑑𝑑𝑑𝑑𝜉𝜉� 𝑐𝑐B,𝑚𝑚,𝑛𝑛(𝜉𝜉)

∞

−∞(16)

and

�𝑔𝑔𝑞𝑞𝑚𝑚,𝑛𝑛B �𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛B† 𝑏𝑏 − 𝑐𝑐𝑞𝑞𝑚𝑚,𝑛𝑛B 𝑏𝑏†�

𝑞𝑞𝑚𝑚,𝑛𝑛B

= 𝑔𝑔𝑚𝑚,𝑛𝑛√𝐿𝐿� 𝑑𝑑𝜉𝜉𝛿𝛿(𝜉𝜉)�𝑐𝑐B,𝑚𝑚,𝑛𝑛† (𝜉𝜉)𝑏𝑏 − 𝑐𝑐B,𝑚𝑚,𝑛𝑛(𝜉𝜉)𝑏𝑏†�

∞

−∞(17)

By substituting Supplementary Equations (8) – (17) into Supplementary Equation (7) we obtain the real-space Hamiltonian as

𝐻𝐻 = ℏ𝜔𝜔0𝑏𝑏†𝑏𝑏 + 𝑖𝑖ℏ𝑣𝑣𝐿𝐿 � �� 𝑑𝑑𝜉𝜉 �𝑐𝑐F,𝑚𝑚,𝑛𝑛† (𝜉𝜉) �−

𝑑𝑑𝑑𝑑𝜉𝜉� 𝑐𝑐F,𝑚𝑚,𝑛𝑛 + 𝑐𝑐B,𝑚𝑚,𝑛𝑛

† (𝜉𝜉)(𝑑𝑑𝑑𝑑𝜉𝜉)𝑐𝑐B,𝑚𝑚,𝑛𝑛(𝜉𝜉)�

∞

−∞

𝑁𝑁

𝑛𝑛=1

𝑀𝑀

𝑚𝑚=1

+ 𝑖𝑖ℏ � �� 𝑑𝑑𝜉𝜉𝑔𝑔𝑚𝑚,𝑛𝑛𝐿𝐿𝛿𝛿(𝜉𝜉) ��𝑐𝑐F,𝑚𝑚,𝑛𝑛† (𝜉𝜉) + 𝑐𝑐B,𝑚𝑚,𝑛𝑛

† (𝜉𝜉)� 𝑏𝑏 − �𝑐𝑐F,𝑚𝑚,𝑛𝑛(𝜉𝜉) + 𝑐𝑐B,𝑚𝑚,𝑛𝑛(𝜉𝜉)� 𝑏𝑏†�∞

−∞

𝑁𝑁

𝑛𝑛=1

𝑀𝑀

𝑚𝑚=1

(18)

Given the real-space Hamiltonian, our next step is to solve the time-independent Schrodinger equation 𝐻𝐻|𝜓𝜓⟩ = ℏ𝜔𝜔𝐪𝐪|𝜓𝜓⟩, where |𝜓𝜓⟩ is the eigenstate. The most general eigenstate can be written in terms of the eigenmodes of the photonic crystal as

|𝜓𝜓⟩ = � ��𝑑𝑑𝜉𝜉�𝜓𝜓F,𝑚𝑚,𝑛𝑛(𝜉𝜉)𝑐𝑐F,𝑚𝑚,𝑛𝑛† (𝜉𝜉) + 𝜓𝜓B,𝑚𝑚,𝑛𝑛(𝜉𝜉)𝑐𝑐B,𝑚𝑚,𝑛𝑛

† (𝜉𝜉)� |0, g⟩𝑁𝑁

𝑛𝑛=1

𝑀𝑀

𝑚𝑚=1

+ 𝐸𝐸𝑏𝑏†|0, g⟩ (19)

where |0, g⟩ indicates that the TLS is in the ground state and 𝐸𝐸 is the excitation amplitude of the TLS. The single-photon wavefunctions 𝜓𝜓F,𝑚𝑚,𝑛𝑛(𝜉𝜉) and 𝜓𝜓B,𝑚𝑚,𝑛𝑛(𝜉𝜉) in the nth channel of the mth Weyl point are given by

𝜓𝜓F,𝑚𝑚,𝑛𝑛(𝜉𝜉) = 𝑢𝑢(𝜉𝜉)𝑒𝑒𝑖𝑖𝑞𝑞𝑚𝑚,𝑛𝑛𝜉𝜉�𝛿𝛿𝑚𝑚𝑚𝑚𝛿𝛿𝑛𝑛𝑛𝑛𝜃𝜃(−𝜉𝜉) + 𝑡𝑡𝑚𝑚.𝑛𝑛𝜃𝜃(𝜉𝜉)� (20)

𝜓𝜓B,𝑚𝑚,𝑛𝑛(𝜉𝜉) = 𝑢𝑢(𝜉𝜉)𝑒𝑒−𝑖𝑖𝑞𝑞𝑚𝑚,𝑛𝑛𝜉𝜉�𝑟𝑟𝑚𝑚,𝑛𝑛𝜃𝜃(−𝜉𝜉)� (21)

where 𝑢𝑢(𝜉𝜉)𝑒𝑒𝑖𝑖𝑞𝑞𝑚𝑚𝜉𝜉 is the eigenmode associated with wave number 𝑞𝑞𝑚𝑚,𝑛𝑛 and 𝜃𝜃(𝜉𝜉) is the Heaviside step function. The Kronecker delta 𝛿𝛿𝑚𝑚𝑚𝑚𝛿𝛿𝑛𝑛𝑛𝑛 indicates that single photons are incident from the in the pth channel of the lth Weyl point. The coefficients 𝑟𝑟𝑚𝑚,𝑛𝑛 and 𝑡𝑡𝑚𝑚,𝑛𝑛 are the amplitudes of the reflected and transmitted photons in the nth channel of the mth Weyl point, respectively. We schematically plot the wavefunctions in Supplementary Figure 1d for clear visualization.

By substituting the eigenstate into the real-space Hamiltonian, we obtain the following set of equations for each channel

−𝑖𝑖𝑣𝑣𝑑𝑑𝑑𝑑𝜉𝜉 �𝛿𝛿𝑚𝑚𝑚𝑚𝛿𝛿𝑛𝑛𝑛𝑛𝑢𝑢(𝜉𝜉)𝜃𝜃(−𝜉𝜉) + 𝑡𝑡𝑚𝑚,𝑛𝑛𝑢𝑢(𝜉𝜉)𝜃𝜃(𝜉𝜉)� + 𝑖𝑖𝑔𝑔𝑚𝑚,𝑛𝑛√𝐿𝐿𝑒𝑒−𝑖𝑖𝑞𝑞𝑚𝑚,𝑛𝑛𝜉𝜉𝐸𝐸 = 0 (22)

𝑖𝑖𝑣𝑣𝑒𝑒𝑖𝑖𝑞𝑞𝑚𝑚,𝑛𝑛𝜉𝜉𝑑𝑑𝑑𝑑𝜉𝜉 𝑟𝑟𝑚𝑚,𝑛𝑛𝑢𝑢(𝜉𝜉)𝜃𝜃(−𝜉𝜉) + 𝑖𝑖𝑔𝑔𝑚𝑚,𝑛𝑛√𝐿𝐿𝑒𝑒𝑖𝑖𝑞𝑞𝑚𝑚,𝑛𝑛𝜉𝜉𝐸𝐸 = 0 (23)

and the following equation for the interaction between the TLS and the propagating waves

𝜔𝜔0𝐸𝐸 − 𝑖𝑖 � �𝑔𝑔𝑚𝑚,𝑛𝑛√𝐿𝐿 �𝑢𝑢(0)�𝛿𝛿𝑚𝑚𝑚𝑚𝛿𝛿𝑛𝑛𝑛𝑛 + 𝑡𝑡𝑚𝑚,𝑛𝑛�

2 +𝑢𝑢(0)𝑟𝑟𝑚𝑚,𝑛𝑛

2 �𝑁𝑁

𝑛𝑛=1

𝑀𝑀

𝑚𝑚=1

= 𝜔𝜔𝐪𝐪𝐸𝐸 (24)

By solving Supplementary Equation (22) and Supplementary Equation (23) we can obtain the coefficients 𝑡𝑡𝑛𝑛 and 𝑟𝑟𝑛𝑛 as

𝑡𝑡𝑚𝑚,𝑛𝑛 =𝑔𝑔𝑚𝑚,𝑛𝑛√𝐿𝐿𝑣𝑣𝑢𝑢(0) 𝐸𝐸 + 𝛿𝛿𝑚𝑚𝑚𝑚𝛿𝛿𝑛𝑛𝑛𝑛 (25)

𝑟𝑟𝑚𝑚,𝑛𝑛 =𝑔𝑔𝑚𝑚,𝑛𝑛√𝐿𝐿𝑣𝑣𝑢𝑢(0) 𝐸𝐸 (26)

By substituting Supplementary Equations (25) – (26) into Supplementary Equation (24), we obtain a simple equation of the excitation amplitude E as

�𝜔𝜔𝐪𝐪 − 𝜔𝜔0 + 𝑖𝑖 � ��𝑔𝑔𝑚𝑚,𝑛𝑛�

2𝐿𝐿𝑣𝑣

𝑁𝑁

𝑛𝑛=1

𝑀𝑀

𝑚𝑚=1

�𝐸𝐸 = −𝑖𝑖𝑔𝑔𝑚𝑚,𝑛𝑛√𝐿𝐿𝑢𝑢(0) (27)

We can then obtain the transmission coefficient 𝑡𝑡𝑚𝑚,𝑛𝑛 and reflection coefficient 𝑡𝑡𝑚𝑚,𝑛𝑛 in the nth channel of the mth Weyl point as

𝑡𝑡𝑚𝑚,𝑛𝑛 = −𝑖𝑖𝑔𝑔𝑚𝑚,𝑛𝑛𝑔𝑔𝑚𝑚,𝑛𝑛𝐿𝐿

𝑣𝑣1

𝜔𝜔𝐪𝐪 − 𝜔𝜔0 + 𝑖𝑖 Γ2+ 𝛿𝛿𝑚𝑚𝑚𝑚𝛿𝛿𝑛𝑛𝑛𝑛 (28)

𝑟𝑟𝑚𝑚,𝑛𝑛 = −𝑖𝑖𝑔𝑔𝑚𝑚,𝑛𝑛𝑔𝑔𝑚𝑚,𝑛𝑛𝐿𝐿

𝑣𝑣1

𝜔𝜔𝐪𝐪 − 𝜔𝜔0 + 𝑖𝑖 Γ2(29)

where Γ = 2∑ ∑ �𝑔𝑔𝑚𝑚,𝑛𝑛�2𝐿𝐿/𝑣𝑣 𝑁𝑁

𝑛𝑛=1𝑀𝑀𝑚𝑚=1 is the spontaneous emission rate of the TLS and the factor of 2

results from the forward and backward propagating waves.

Thus, the scattering cross section can be calculated as

𝜎𝜎�𝜔𝜔𝐪𝐪, 𝑞𝑞𝑚𝑚,𝑛𝑛� =∑ ∑ �|𝑡𝑡𝑚𝑚,𝑛𝑛 − 𝛿𝛿𝑚𝑚𝑚𝑚𝛿𝛿𝑛𝑛𝑛𝑛|2 + |𝑟𝑟𝑚𝑚,𝑛𝑛|2�𝑁𝑁

𝑛𝑛=1𝑀𝑀𝑚𝑚=1

1/𝐿𝐿2

=2�𝑔𝑔𝑚𝑚,𝑛𝑛�

2𝐿𝐿3

𝑣𝑣∑ ∑ �𝑔𝑔𝑚𝑚,𝑛𝑛�

2𝐿𝐿/𝑣𝑣𝑁𝑁𝑛𝑛=1

𝑀𝑀𝑚𝑚=1

(𝜔𝜔0 − 𝜔𝜔𝐪𝐪)2 + Γ24

= �𝑔𝑔𝑚𝑚,𝑛𝑛�

2𝐿𝐿3

𝑣𝑣Γ

�𝜔𝜔0 − 𝜔𝜔𝐪𝐪�2 + Γ2

4

(30)

We can further normalize the coupling coefficient �𝑔𝑔𝑚𝑚,𝑛𝑛�2𝐿𝐿/𝑣𝑣 by its maximum (𝑔𝑔0)2𝐿𝐿/𝑣𝑣, and define an

angular factor 𝑓𝑓𝑚𝑚,𝑛𝑛 = �𝑔𝑔𝑚𝑚,𝑛𝑛/𝑔𝑔0�2. Supplementary Equation (30) then can be rewritten to an expression

similar to the well-known Breit-Wigner formula as

𝜎𝜎�𝜔𝜔𝐪𝐪, 𝑞𝑞𝑚𝑚,𝑛𝑛� =2𝑓𝑓𝑚𝑚,𝑛𝑛𝐿𝐿2

∑ ∑ 𝑓𝑓𝑚𝑚,𝑛𝑛 𝑁𝑁𝑛𝑛=1

𝑀𝑀𝑚𝑚=1

Γ24

�𝜔𝜔0 − 𝜔𝜔𝒒𝒒�2 + Γ2

4

=4𝑓𝑓𝑚𝑚,𝑛𝑛𝐿𝐿2(∆𝑞𝑞)2

|𝑞𝑞𝑚𝑚.𝑛𝑛|2 ∑ ∮ cos𝜃𝜃𝑚𝑚.𝑛𝑛 𝑓𝑓𝑚𝑚,𝑛𝑛𝑑𝑑Ω𝑚𝑚,𝑛𝑛𝑀𝑀𝑚𝑚=1

Γ24

�𝜔𝜔0 − 𝜔𝜔𝐪𝐪�2 + Γ2

4

=𝑓𝑓𝑚𝑚,𝑛𝑛ℱ

16𝜋𝜋2𝑣𝑣2

(𝜔𝜔0 − 𝜔𝜔𝑊𝑊𝑊𝑊𝑦𝑦𝑚𝑚)2

Γ24

�𝜔𝜔0 − 𝜔𝜔𝐪𝐪�2 + Γ2

4

(31)

where Ω𝑚𝑚,𝑛𝑛 is the solid angle associated with 𝑞𝑞𝑚𝑚,𝑛𝑛, ℱ = ∑ ∮ cos 𝜃𝜃𝑚𝑚.𝑛𝑛 𝑓𝑓𝑚𝑚,𝑛𝑛𝑑𝑑Ω𝑚𝑚,𝑛𝑛𝑀𝑀𝑚𝑚=1 and 𝑓𝑓𝑚𝑚,𝑛𝑛/ℱ indicates

the angular distribution of the scattering cross section. The prefactor 16𝑣𝑣2/(𝜔𝜔0 − 𝜔𝜔Weyl)2 ultimately determines the maximum cross section, which agrees with the conclusion in the main text. Note we have assumed that the isosurface around each Weyl point is isotropic throughout the derivation, i.e. |𝑞𝑞𝑚𝑚.𝑛𝑛| =�𝜔𝜔𝑞𝑞𝑚𝑚.𝑛𝑛 − 𝜔𝜔Weyl�/𝑣𝑣. For anisotropic Weyl point, the scattering cross section has the same form and only differs from the isotropic case by a constant.

Supplementary Note 3: Frequency dependence of the spectrum of the average cross section

In the main text, we focus on the maximum of the average cross section when 𝜔𝜔𝐪𝐪 = 𝜔𝜔0 and briefly discuss the spectrum of the average cross section in Fig. 4. Here we will discuss more details about the frequency dependence of the spectrum of the average cross section.

The average cross section can be directly calculated by using Supplementary Equation (30). To simplify the discussion, here we will also assume that the isosurface around each Weyl point is identical and isotropic, i.e., |𝑞𝑞𝑚𝑚.𝑛𝑛| = �𝜔𝜔𝑞𝑞𝑚𝑚.𝑛𝑛 − 𝜔𝜔Weyl�/𝑣𝑣 . Recall that (𝑔𝑔0)2 = 𝑑𝑑2𝜔𝜔0/2ℏ𝜖𝜖0𝐿𝐿3 and 𝑑𝑑 is the dipole moment of the TLS. Thereby, the spontaneous emission rate Γ can also be calculated as

Γ = 2 � ��𝑔𝑔𝑚𝑚,𝑛𝑛�

2𝐿𝐿𝑣𝑣

𝑁𝑁

𝑛𝑛=1

𝑀𝑀

𝑚𝑚=1

=(𝑔𝑔0)2𝐿𝐿3�𝜔𝜔0 − 𝜔𝜔Weyl�

2

4𝜋𝜋2𝑣𝑣3 � � cos𝜃𝜃𝑚𝑚.𝑛𝑛 𝑓𝑓𝑚𝑚,𝑛𝑛𝑑𝑑Ω𝑚𝑚,𝑛𝑛

𝑀𝑀

𝑚𝑚=1

𝑑𝑑2𝜔𝜔0�𝜔𝜔0 − 𝜔𝜔Weyl�2

8ℏ𝜖𝜖0𝜋𝜋2𝑣𝑣3� � cos𝜃𝜃𝑚𝑚.𝑛𝑛 𝑓𝑓𝑚𝑚,𝑛𝑛𝑑𝑑Ω𝑚𝑚,𝑛𝑛

𝑀𝑀

𝑚𝑚=1

= 𝓅𝓅�𝜔𝜔0 − 𝜔𝜔Weyl�2 (32)

where Ω𝑚𝑚,𝑛𝑛 is the solid angle associated with wave number 𝑞𝑞𝑚𝑚,𝑛𝑛 . For simplification, we define a coefficient 𝓅𝓅 = 𝑑𝑑2𝜔𝜔0 ∑ ∮ cos 𝜃𝜃𝑚𝑚.𝑛𝑛 𝑓𝑓𝑚𝑚,𝑛𝑛𝑑𝑑Ω𝑚𝑚,𝑛𝑛

𝑀𝑀𝑚𝑚=1 /8ℏ𝜖𝜖0𝜋𝜋2𝑣𝑣3. Around the Weyl point, the coefficient 𝓅𝓅

does not vary significantly with the frequency. As a result, the spontaneous emission rate Γ scales as

�𝜔𝜔0 − 𝜔𝜔Weyl�2.

Next, we calculate the average cross section by integrating the cross section on the isosurfaces and then dividing the integration by the area of the isosurfaces. The spectrum of the average cross section then is given by

𝜎𝜎��𝜔𝜔𝐪𝐪� =(𝑔𝑔0)2𝐿𝐿3

𝑣𝑣∑ ∮ cos 𝜃𝜃𝑚𝑚.𝑛𝑛 𝑓𝑓𝑚𝑚,𝑛𝑛𝑑𝑑Ω𝑚𝑚,𝑛𝑛𝑀𝑀𝑚𝑚=1

∑ ∮𝑑𝑑Ω𝑚𝑚,𝑛𝑛𝑀𝑀𝑚𝑚=1

Γ

�𝜔𝜔0 − 𝜔𝜔𝐪𝐪�2 + Γ2

4

= 𝑑𝑑2𝜔𝜔0

8𝑀𝑀ℏ𝜖𝜖0𝜋𝜋𝑣𝑣� � cos 𝜃𝜃𝑚𝑚.𝑛𝑛 𝑓𝑓𝑚𝑚,𝑛𝑛𝑑𝑑Ω𝑚𝑚,𝑛𝑛

𝑀𝑀

𝑚𝑚=1

Γ

�𝜔𝜔0 − 𝜔𝜔𝐪𝐪�2 + Γ2

4

= 𝜋𝜋𝓅𝓅2𝑣𝑣2

𝑀𝑀�𝜔𝜔0 − 𝜔𝜔Weyl�

2

�𝜔𝜔0 − 𝜔𝜔𝐪𝐪�2 +

𝓅𝓅2�𝜔𝜔0 − 𝜔𝜔Weyl�4

4

(33)

Note the prefactor 𝜋𝜋𝓅𝓅2𝑣𝑣2/𝑀𝑀 does not vary significantly with the frequency around the Weyl point. On

resonance, i.e., 𝜔𝜔𝐪𝐪 = 𝜔𝜔0, the resonant average cross section scales as 1/�𝜔𝜔0 − 𝜔𝜔Weyl�2 and increases

drastically as 𝜔𝜔0 → 𝜔𝜔Weyl . In great contrast, the cross section of off-resonance frequencies scales as

�𝜔𝜔0 − 𝜔𝜔Weyl�2/�𝜔𝜔0 − 𝜔𝜔𝐪𝐪�

2 and is strongly suppressed around the Weyl point.

Meanwhile, it’s clear to see that the integration of the average cross section ∫ 𝜎𝜎��𝜔𝜔𝐪𝐪�𝑑𝑑𝜔𝜔𝐪𝐪∞−∞ is roughly a

constant, which can be shown as

� 𝜎𝜎��𝜔𝜔𝒒𝒒�𝑑𝑑𝜔𝜔𝒒𝒒∞

−∞= � 𝑑𝑑𝜔𝜔𝒒𝒒

∞

−∞

𝜋𝜋𝓅𝓅2𝑣𝑣2

𝑀𝑀�𝜔𝜔0 − 𝜔𝜔Weyl�

2

�𝜔𝜔𝒒𝒒 − 𝜔𝜔0�2 +

𝓅𝓅2�𝜔𝜔0 − 𝜔𝜔Weyl�4

4

=2𝓅𝓅𝜋𝜋2𝑣𝑣2

𝑀𝑀(34)

where 2𝓅𝓅𝜋𝜋2𝑣𝑣2/𝑀𝑀 remains around the same around the Weyl point.

Supplementary Note 4: Resonant scattering in Dirac systems

The conservation law of resonant scattering can be also extended to two-dimensional (2D) space. In 2D space, the conical dispersion forms a Dirac point, which is the 2D analogy of the Weyl point. In this section, we will use a 2D Dirac photonic crystal as an example and demonstrate the diverging resonant scattering cross section around the Dirac point.

We consider a triangular lattice of dielectric rods (grey circle in Supplementary Figure 2a), which has a dielectric constant of 𝜖𝜖 = 12. The radius of the rods is 0.3𝑎𝑎, where a is the lattice constant. We then solve the band structure of this Dirac photonic crystal by using MPB7. We obtain 6 Dirac points in the band structure for the transverse electric (TE) bands at 𝜔𝜔 = 0.462 (2𝜋𝜋𝑐𝑐/𝑎𝑎). To visualize the momentum space in 2D space, we plot the isofrequency contour for 𝜔𝜔 = 0.461 (2𝜋𝜋𝑐𝑐/𝑎𝑎) in Supplementary Figure 2b. For each Dirac point, the isofrequency contour is almost a circle.

Supplementary Figure 2: Resonant scattering in Dirac systems. a Schematic of the 2D Dirac photonic crystal. b Isofrequency contour of the Dirac photonic crystal at 𝜔𝜔 = 0.461 (2𝜋𝜋𝑐𝑐/𝑎𝑎). 6 Dirac points are obtained at 𝜔𝜔 = 0.462 (2𝜋𝜋𝑐𝑐/𝑎𝑎) for the TE bands. The isofrequency contour is almost a circle. c The resonant cross section of the TLS for different locations. The resonant frequency of the TLS is 𝜔𝜔0 = 0.461 (2𝜋𝜋𝑐𝑐/𝑎𝑎). The integration in momentum space always leads to the same constant. Positions A-B-C-A are labelled in a. Examples of 𝜎𝜎(𝐪𝐪) at two different positions are plotted as insets. d Diverging

average resonant cross section is realized around the Dirac frequency. The results from quantum scattering simulation (red circles) agree well with the prediction based on the band structure (blue dashed line). The cross section is normalized by the average cross section in free space 𝜎𝜎�0 = 2𝜆𝜆/𝜋𝜋. The black dashed line indicates the Dirac frequency.

Next, we numerically verify the conservation law of resonant scattering in Dirac systems. As an example, we consider a TLS with a transition frequency of 𝜔𝜔0 = 0.461 (2𝜋𝜋𝑐𝑐/𝑎𝑎). The dipole moment of the TLS is in the x direction. The calculated cross sections 𝜎𝜎(𝐪𝐪) of the TLS at two different positions are plotted as insets of Supplementary Figure 2c. Similar to the cross section in Weyl photonic crystals, it also strongly depends on the incident wavevector 𝐪𝐪 = 𝐤𝐤 − 𝐤𝐤Dirac and varies significantly at different locations.

The conversation law of resonant cross section in 2D space is slightly different from that in 3D space. Instead of integrating over the isosurface, we now integrate the resonant cross section over the isofrequency contour and the conservation law becomes

�𝜎𝜎(𝐪𝐪)𝑑𝑑𝑑𝑑 = 8𝜋𝜋 (35)

To confirm the above conservation law, we integrate the resonant cross section for a TLS at 16 different locations and plot the results in Supplementary Figure 2c. As expected, the integration ∫𝜎𝜎(𝐪𝐪)𝑑𝑑𝑑𝑑 always results in the same constant of 8𝜋𝜋.

We then sweep the transition frequency of the TLS to across the Dirac point, which is indicated by black dashed line in Supplementary Figure 2d. To better demonstrate the enhancement, we also define an average cross section 𝜎𝜎� = ∫𝜎𝜎(𝐪𝐪)𝑑𝑑𝑑𝑑 /𝐶𝐶, where C is the total circumference of the isofrequency contours. The calculated average cross section is plotted as red circles in Supplementary Figure 2d. It’s enhanced by almost three orders of magnitude compared to 𝜎𝜎�0 = 2𝜆𝜆/𝜋𝜋, the average cross section in free space. At the Dirac frequency, the resonant cross section diverges.

Supplementary Note 5. Proof of suppressed Rayleigh (non-resonant) scattering in Weyl systems

The suppressed Rayleigh scattering in Weyl system can be directly proved by using perturbation theory and Born approximation. In this section, we will describe the mathematical proof in detail.

Under the framework of the first-order Born approximation8, the scattering amplitude 𝑓𝑓(𝐤𝐤s,𝐤𝐤inc)for a Rayleigh scatterer with a weak scattering potential 𝐕𝐕(𝐫𝐫) is given by

𝑓𝑓(𝐤𝐤s,𝐤𝐤inc) = �𝑑𝑑3𝐫𝐫′𝐮𝐮𝐤𝐤s �𝐫𝐫′�𝐕𝐕 �𝐫𝐫′�𝐮𝐮𝐤𝐤inc �𝐫𝐫

′� 𝑒𝑒𝑖𝑖(𝐤𝐤inc−𝐤𝐤s)∙𝐫𝐫′ (36)

Here 𝐮𝐮𝐤𝐤𝒆𝒆𝑖𝑖𝐤𝐤∙𝐫𝐫 is the eigenmode of the surrounding medium associated with wave vector 𝐤𝐤. 𝐤𝐤s and 𝐤𝐤inc are the wave vectors of the scattered and incident eigenmodes, respectively. For simplicity, we assume the scatterer is very small compared to the incident wavelength and can be considered as a point scatterer located at 𝐫𝐫0. The scattering potential then can be written as 𝐕𝐕(𝐫𝐫) = 𝜔𝜔𝐕𝐕𝛿𝛿(𝐫𝐫 − 𝐫𝐫0). Note here 𝐕𝐕 is a tensor containing the dielectric constant of the scatterer. Thus, the scattering amplitude becomes

𝑓𝑓(𝐤𝐤s,𝐤𝐤inc) = 𝜔𝜔𝐮𝐮𝐤𝐤s(𝐫𝐫0)𝐕𝐕𝐮𝐮𝐤𝐤inc(𝐫𝐫0)𝑒𝑒𝑖𝑖(𝐤𝐤inc−𝐤𝐤s)∙𝐫𝐫0 (37)

The total scattering cross section then can be calculated as

𝜎𝜎(𝜔𝜔,𝐤𝐤inc) = �|𝑓𝑓(𝐤𝐤s,𝐤𝐤inc)|2𝐤𝐤s

= 𝜔𝜔2��𝐮𝐮𝐤𝐤s(𝐫𝐫0)𝐕𝐕𝐮𝐮𝐤𝐤inc(𝐫𝐫0)�2

𝐤𝐤s

(38)

The summation over 𝐤𝐤s can be readily converted to a surface integral over the isosurface given by 𝜔𝜔(𝐤𝐤s) = 𝜔𝜔, and Supplementary Equation (38) further becomes

𝜎𝜎(𝜔𝜔,𝐤𝐤inc) = 𝜔𝜔2 � 𝑑𝑑2𝐤𝐤s 𝑆𝑆:𝜔𝜔(𝐤𝐤𝐬𝐬)=𝜔𝜔

�𝐮𝐮𝐤𝐤s(𝐫𝐫0)𝐕𝐕𝐮𝐮𝐤𝐤inc(𝐫𝐫0)�2 ∝ 𝜔𝜔2𝑆𝑆 (39)

where 𝑆𝑆 is the area of the isosurface. In free space, the area of the isosurface S scales 𝑆𝑆~𝜔𝜔2. Consequently, the Rayleigh scattering cross section scales as 𝜎𝜎(𝜔𝜔,𝐤𝐤inc)~𝜔𝜔4. In great contrast, the area of the isosurface S scales 𝑆𝑆~∆𝜔𝜔2 = (𝜔𝜔 − 𝜔𝜔Weyl)2 and thus the Rayleigh scattering cross section scales as 𝜎𝜎(𝜔𝜔,𝐤𝐤inc)~𝜔𝜔2(𝜔𝜔 − 𝜔𝜔Weyl)2 in Weyl systems.

Supplementary Note 6. Numerical calculation of Rayleigh (non-resonant) scattering in Weyl systems

The Rayleigh scattering cross section can be rigorously calculated by using the normal-mode expansion of the dipole field9–11 that radiated by the scatterer. In this section, we will briefly describe the normal-mode expansion method that we implemented to numerically calculate the Rayleigh scattering cross section.

In a nonmagnetic medium with a dielectric constant of 𝜖𝜖(𝐫𝐫), the Maxwell’s equations are given by

∇ × 𝐄𝐄 = −𝜇𝜇0𝜕𝜕𝐇𝐇𝜕𝜕𝑡𝑡

(40a)

∇ × 𝐇𝐇 = −𝜖𝜖(𝐫𝐫)𝜖𝜖0𝜕𝜕𝐄𝐄𝜕𝜕𝑡𝑡 + 𝐉𝐉 (40b)

∇ ∙ (𝜖𝜖(𝐫𝐫)𝜖𝜖0𝐄𝐄) = 0 (40c)

∇ ∙ 𝐇𝐇 = 0 (40d)

where 𝜇𝜇0 is vacuum permeability. The electric (magnetic) field is denoted by 𝐄𝐄 (𝐇𝐇). An electromagnetic field is produced by a current density 𝐉𝐉. Since the charge density is zero, the scalar potential is zero. The electric and magnetic fields can be written in terms of vector potential 𝐀𝐀 as

𝐄𝐄 = −∂𝐀𝐀∂𝑡𝑡

(41a)

𝐇𝐇 =1𝜇𝜇0∇ × 𝐀𝐀 (41b)

Thus, the wave equation for the vector potential 𝐀𝐀 is given by

∇ × ∇ × 𝐀𝐀 +𝜖𝜖(𝐫𝐫)𝑐𝑐2

∂2𝐀𝐀∂2𝑡𝑡 = 𝜇𝜇0𝐉𝐉 (42)

where 𝑐𝑐 is the speed of light.

The homogeneous solutions to the above wave equation can be written as a superposition of the eigenmodes 𝐀𝐀𝐤𝐤(𝐫𝐫) = 𝐮𝐮𝐤𝐤(𝐫𝐫)𝑒𝑒𝑖𝑖𝐤𝐤∙𝐫𝐫 of the photonic crystal, which satisfy the homogenous wave equation

∇ × ∇ × 𝐀𝐀𝐤𝐤(𝐫𝐫) +𝜔𝜔𝐤𝐤2

𝑐𝑐2 𝜖𝜖(𝐫𝐫)𝐀𝐀𝐤𝐤(𝐫𝐫) = 0 (43)

The eigenmodes 𝐀𝐀𝐤𝐤(𝐫𝐫) also have to satisfy the orthogonalization, normalization and closure conditions given by

�𝑑𝑑3𝐫𝐫 𝜖𝜖(𝐫𝐫)𝐀𝐀𝐤𝐤(𝐫𝐫)𝐀𝐀𝐤𝐤′∗ (𝐫𝐫) = (2𝜋𝜋)3𝐿𝐿3𝛿𝛿(𝐤𝐤 − 𝐤𝐤′) (44)

Here 𝐿𝐿3 is the normalization volume.

We then consider a generic Rayleigh scatterer embedded in such a medium. During the Rayleigh scattering process, the incident wave induces an oscillating dipole moment 𝐝𝐝 = 𝛼𝛼𝐄𝐄inc(𝐫𝐫0) in the scatterer. Here 𝛼𝛼 is the polarizability of the scatterer, and 𝐄𝐄inc(𝐫𝐫0) is the local incident electric field at the position of the scatterer. Since the scatterer is quite small compared to the wavelength, the external current source 𝐉𝐉 can be written as

𝐉𝐉(𝐫𝐫, 𝑡𝑡) = −𝑖𝑖𝜔𝜔0𝐝𝐝𝛿𝛿(𝐫𝐫 − 𝐫𝐫0)𝑒𝑒−𝑖𝑖𝜔𝜔0𝑡𝑡 (45)

where 𝐫𝐫0 is the position of the scatterer. The vector potential 𝐀𝐀(𝐫𝐫, t) then can be obtained by solving Supplementary Equation (42) in terms of 𝐉𝐉 via the dyadic Green’s function 𝐆𝐆 as

𝐀𝐀(𝐫𝐫, 𝑡𝑡) = � 𝑑𝑑𝑡𝑡′∞

−∞�𝑑𝑑3𝐫𝐫′ 𝐉𝐉 �𝐫𝐫′, 𝑡𝑡�𝐆𝐆 �𝐫𝐫, 𝑡𝑡; 𝐫𝐫′, 𝑡𝑡′� (46)

The exact expression of the dyadic Green’s function 𝐆𝐆 can be found in Supplementary Reference [8] and we will not discuss the technical details here. Given the vector potential 𝐀𝐀(𝐫𝐫, 𝑡𝑡), one can show that the time-average radiation power from the induced dipole moment 𝐝𝐝 is given by11

𝑃𝑃(𝜔𝜔) =𝜋𝜋𝜔𝜔2

4𝜖𝜖0� 𝑑𝑑2𝐤𝐤

𝜔𝜔(𝐤𝐤)=𝜔𝜔

|𝐮𝐮𝐤𝐤(𝐫𝐫0) ∙ 𝐝𝐝|2

𝑣𝑣𝑔𝑔,𝐤𝐤(47)

Here 𝑣𝑣𝑔𝑔,𝐤𝐤 is the group velocity associated with wave vector 𝐤𝐤. Then the Rayleigh scattering cross section can be calculated as

𝜎𝜎(𝜔𝜔,𝐤𝐤inc) =𝑃𝑃(𝜔𝜔)

12𝜇𝜇0

𝐮𝐮𝐤𝐤inc(𝐫𝐫0) × �∇ × 𝐮𝐮𝐤𝐤inc∗ (𝐫𝐫0)�

(48)

Here we assume the incident wave is one of the eigenmodes associated with wave vector 𝐤𝐤inc . The denominator in Supplementary Equation (48) indicates the Poynting vector of the incident wave at the location of the scatterer, which usually does not vary much with the frequency. On the other hand, Supplementary Equation (47) clearly indicates that the radiation power 𝑃𝑃(𝜔𝜔) is proportional to the area of the isosurface S as 𝑃𝑃(𝜔𝜔) ~𝜔𝜔2𝑆𝑆 . Consequently, the Rayleigh scattering cross scales as 𝜎𝜎~𝜔𝜔2𝑆𝑆, which is consistent with Supplementary Equation (39) that we derived by using perturbation theory.

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